Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{-3z^2 - 30z - 63}{-8z^2 + 8z + 96}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {-3(z^2 + 10z + 21)} {-8(z^2 - z - 12)} $ $ t = \dfrac{3}{8} \cdot \dfrac{z^2 + 10z + 21}{z^2 - z - 12} $ Next factor the numerator and denominator. $ t = \dfrac{3}{8} \cdot \dfrac{(z + 3)(z + 7)}{(z + 3)(z - 4)}$ Assuming $z \neq -3$ , we can cancel the $z + 3$ $ t = \dfrac{3}{8} \cdot \dfrac{z + 7}{z - 4}$ Therefore: $ t = \dfrac{ 3(z + 7)}{ 8(z - 4)}$, $z \neq -3$